The main motivation of this paper is
to investigate the interaction structure between the business sectors in complex financial systems.
Especially, we extract the local interactions between the business sectors with the random matrix decomposition.
The methodology is to combine the RMT theory with various methods
in complex networks, such as the MST tree and PMFG graph, and the theoretical information method (Infomap).
In a complex network, the community structure is the gathering of
nodes into communities with a very high density of internal edges,
including a relatively high density of edges between related communities.
Exploring the community structure with the concepts and methods in statistical mechanics is a very important
topic in the field of complex networks [1]}, [2]}, [3]}, [4]}.
Our study in this paper will provide a deeper understanding on the communities, i.e., the business sectors,
the interactions between the business sectors, and how this business-sector structure emerges
from the correlations between individual stocks.
In particular, with the random matrix decomposition based on the RMT theory,
we clarify that the local interactions between the business
sectors are mainly contained in the sector mode.
In the sector mode,
the average correlation inside the sectors is positive,
while that between the sectors is negative.
Further, we study the time evolution of the interaction structure.
We discover that the interaction structure of the sector mode varies dramatically with time, while that of the full cross-correlation matrix does not.
Particularly, the interaction structure of the business sectors in the sector mode is simple during a financial bubble or crisis,
and it is dominated by specific business sectors, which are associated with the corresponding economic situation.
